(see also Chern-Weil theory, parameterized homotopy theory)
The Adams conjecture is a statement about triviality of spherical fibrations associated to certain formal differences of vector bundles (K-theory classes) via the J-homomorphism. The conjecture was stated in (Adams 63, conjecture 1.2), for vector bundles of rank up to two over a finite CW-complex, which was proven in (Adams 63, theorem 1.4). A general proof was then given in (Quillen 71).
The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the image of the J-homomorphism.
Let $X$ be (the homotopy type of) a topological space. For $V \;\colon\; X \longrightarrow B O$ classifying a real vector bundle on $X$, the corresponding spherical fibration is classified by the composite
with the delooped J-homomorphism. This descends to a map from topological K-theory to spherical fibrations.
Now for $L$ a line bundle on some $X$ and for non-vanishing $k \in \mathbb{Z}$, John Adams observed that the spherical fibration associated with the difference $L^{\otimes k} - L \in K O(X)$ has the property that some $k$-fold multiple of it has trivial spherical fibration, hence that there is $N \in \mathbb{N}$ for which
Noticing that $L \mapsto L^{\otimes^k} = \Psi^k(L)$ is the $k$th Adams operation on K-theory applied to the line bundle $L$, John Adams then conjectured that the above is true for all vector bundles $V$ in the form
The conjecture originates in:
Textbook accounts:
Review:
Akhil Mathew, The Adams conjecture I (web)
Michael Hopkins (notes by Akhil Mathew), Lecture 30 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)
The proof of the Adams conjecture is originally due to
The proof using algebraic geometry is due to
Yet another proof via Becker-Gottlieb transfer is due to
The generalization to equivariant cohomology (equivariant K-theory) is discussed in
Tammo tom Dieck, theorem 11.3.8 in Transformation Groups and Representation Theory Lecture Notes in Mathematics 766 Springer 1979
Z. Fiedorowicz, H. Hauschild, Peter May, theorem 0.4 of Equivariant algebraic K-theory, Equivariant algebraic K-theory, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (pdf)
Henning Hauschild, Stefan Waner, theorem 0.1 of The equivariant Dold theorem mod $k$ and the Adams conjecture, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (euclid:1256065410)
Kuzuhisa Shimakawa, Note on the equivariant $K$-theory spectrum, Publ. RIMS, Kyoto Univ. 29 (1993), 449-453 (pdf, doi)
Christopher French, theorem 2.4 in The equivariant $J$–homomorphism for finite groups at certain primes, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (euclid:1513797069)